Entanglement properties of topological color codes

Mehdi Kargarian

Published 2008-09-25, updated 2008-12-07Version 5

The entanglement properties of a class of topological stabilizer states, the so called \emph{topological color codes} defined on a two-dimensional lattice or \emph{2-colex}, are calculated. The topological entropy is used to measure the entanglement of different bipartitions of the 2-colex. The dependency of the ground state degeneracy on the genus of the surface shows that the color code can support a topological order, and the contribution of the color in its structure makes it interesting to compare with the Kitaev's toric code. While a qubit is maximally entangled with rest of the system, two qubits are no longer entangled showing that the color code is genuinely multipartite entangled. For a convex region, it is found that entanglement entropy depends only on the degrees of freedom living on the boundary of two subsystems. The boundary scaling of entropy is supplemented with a topological subleading term which for a color code defined on a compact surface is twice than the toric code. From the entanglement entropy we construct a set of bipartitions in which the diverging term arising from the boundary term is washed out, and the remaining non-vanishing term will have a topological nature. Besides the color code on the compact surface, we also analyze the entanglement properties of a version of color code with border, i.e \emph{triangular color code}.